Initial Successive coefficients of Inverse functions of certain classes of univalent functions

TL;DR

This paper investigates bounds on the differences of successive coefficients of inverse functions for certain classes of univalent functions, providing sharp estimates for specific subclasses within the unit disk.

Contribution

It establishes sharp bounds on the differences of successive inverse coefficients for subclasses of univalent functions, extending understanding of their coefficient behavior.

Findings

Sharp bounds for |A_2| - |A_1| and |A_3| - |A_2| are derived.

Results apply to specific subclasses of univalent functions.

The bounds are proven to be optimal.

Abstract

We consider functions of the type $f(z)=z+a_2z^2+a_3z^3+\cdots$ from a family of all analytic and univalent functions in the unit disk. Let $F$ be the inverse function of $f$, given by $F(z)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some $|w|\le r_0(f)$. In this paper, we find the sharp bounds of $\big | |A_{n+1}|-|A_n|\big |$, for $n=1,\,2$, for some subclasses of univalent functions.